Chapter 9.3, Problem 52E

a.

To determine

To compare the rates of growth of $x^5\text{ and }{x}^2\text{ as }x\to \infty$

Answer to Problem 52E

On comparison of rates it is found that $x^5$ grows faster than $x^2$

Explanation of Solution

Given information:

The rates of growth of $x^5$ and $x^2$ can be find out as,

  $\underset{x\to \infty }{\lim }\frac{x^5}{x^2}=\underset{x\to \infty }{\lim }{x}^3=\infty$

Therefore, the rates of growth is obtained $\infty$ , so according to the definition of rate of growth it is concluded that $x^5$ grows faster than $x^2$ as $x\to \infty$

b.

To determine

To compare the rate of growth of $5x^3\text{ and 2}{x}^3\text{ as }x\to \infty$

Answer to Problem 52E

On comparison of rates it is found that $5x^3$ and $\text{2}{x}^3$ grows at the same rate

Explanation of Solution

Given information:

The rates of growth of $5x^3$ and $\text{2}{x}^3$ can be find out as,

  $\underset{x\to \infty }{\lim }\frac{5x^3}{2x^3}=\underset{x\to \infty }{\lim }\frac{5}{2}=\frac{5}{2}$

Therefore, the rate of growth is obtained as $\frac{5}{2}$ which is not equal to zero, so according to the definition of rate of growth it is concluded that $5x^3$ and $\text{2}{x}^3$ grows at the same rate as $x\to \infty$

c.

To determine

To give conclusion about $m\text{ and }n$ if $x^m$ grows faster than $x^n$ as $x\to \infty$

Answer to Problem 52E

Conclusion $-$ If $x^m$ grows faster than $x^n$ , $\underset{x\to \infty }{\lim }{x}^{m-n}=\infty$ which means $m-n>0\text{ or }m>n$

Explanation of Solution

Given information:

  $\underset{x\to \infty }{\lim }\frac{x^m}{x^n}=\underset{x\to \infty }{\lim }{x}^{m-n}$

Therefore, if $x^m$ grows faster than $x^n$ , $\underset{x\to \infty }{\lim }{x}^{m-n}=\infty$ which means $m-n>0\text{ or }m>n$

d.

To determine

To give conclusion about $m\text{ and }n$ if $x^m$ grows at the same rate as $x^n$ as $x\to \infty$

Answer to Problem 52E

Conclusion $-$ If $x^m$ and $x^n$ grow at the same rate, $\underset{x\to \infty }{\lim }{x}^{m-n}=$ a nonzero constant, which means $m-n=0\text{ or }m=n$

Explanation of Solution

Given information:

  $\underset{x\to \infty }{\lim }\frac{x^m}{x^n}=\underset{x\to \infty }{\lim }{x}^{m-n}$

Therefore, if $x^m$ and $x^n$ grow at the same rate, $\underset{x\to \infty }{\lim }{x}^{m-n}=$ a nonzero constant, which means $m-n=0\text{ or }m=n$

e.

To determine

To give the conclusion about the degrees of $g\left(x\right)\text{ and }f\left(x\right)$ if $g\left(x\right)$ grows faster than $f\left(x\right)$ as $x\to \infty$

Answer to Problem 52E

Conclusion $-$ The degree of $g\left(x\right)$ grow faster than $f\left(x\right)$ as $x\to \infty$ if $\underset{x\to \infty }{\lim }\frac{g\left(x\right)}{f\left(x\right)}$ must be equal to $\infty$

Explanation of Solution

Given information:

  $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0,a_n>0$

And

  $g\left(x\right)=b_m{x}^m+b_{m-1}{x}^{m-1}+\cdots +b_{1}x+b_0,b_m>0$

  $\begin{array}{l}\underset{x\to \infty }{\lim }\frac{g\left(x\right)}{f\left(x\right)}=\frac{b_m{x}^m+b_{m-1}{x}^{m-1}+\cdots +b_{1}x+b_0,b_m>0}{a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0,a_n>0}\\ \underset{x\to \infty }{\lim }\frac{g\left(x\right)}{f\left(x\right)}=\frac{b_m{x}^{m-1}+b_{m-1}{x}^{m-2}+\cdots b_1}{a_n{x}^{n-1}+a_{n-1}{x}^{n-2}+\cdots a_1}\end{array}$

Therefore, if $g\left(x\right)$ grow faster than $f\left(x\right)$ as $x\to \infty$ then $\underset{x\to \infty }{\lim }\frac{g\left(x\right)}{f\left(x\right)}$ must be equal to $\infty$

f.

To determine

To give the conclusion about the degrees of $g\left(x\right)\text{ and }f\left(x\right)$ if $g\left(x\right)$ grows at the same rate as $f\left(x\right)$ as $x\to \infty$

Answer to Problem 52E

Conclusion $-$ The degree of $g\left(x\right)$ grow at the same rate as $f\left(x\right)$ as $x\to \infty$ if $\underset{x\to \infty }{\lim }\frac{g\left(x\right)}{f\left(x\right)}$ = nonzero constant.

Explanation of Solution

Given information:

  $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0,a_n>0$

And

  $g\left(x\right)=b_m{x}^m+b_{m-1}{x}^{m-1}+\cdots +b_{1}x+b_0,b_m>0$

  $\begin{array}{l}\underset{x\to \infty }{\lim }\frac{g\left(x\right)}{f\left(x\right)}=\frac{b_m{x}^m+b_{m-1}{x}^{m-1}+\cdots +b_{1}x+b_0,b_m>0}{a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0,a_n>0}\\ \underset{x\to \infty }{\lim }\frac{g\left(x\right)}{f\left(x\right)}=\frac{b_m{x}^{m-1}+b_{m-1}{x}^{m-2}+\cdots b_1}{a_n{x}^{n-1}+a_{n-1}{x}^{n-2}+\cdots a_1}\end{array}$

Therefore, if $g\left(x\right)$ grows at the same rate as $f\left(x\right)$ as $x\to \infty$ then $\underset{x\to \infty }{\lim }\frac{g\left(x\right)}{f\left(x\right)}$ = nonzero constant